Let n be a non-negative integer.
Find the non-negative integers a, b,
c, d such that
a2+b2+c2+d2=7*4n
DEFDBL A-Z
FOR n = 0 TO 10
PRINT n
rhs = INT(7 * 4 ^ n + .5)
FOR a = 0 TO SQR(rhs)
FOR b = a TO SQR(rhs - a * a)
FOR c = b TO SQR(rhs - a * a - b * b)
rm = rhs - a * a - b * b - c * c
d = INT(SQR(rm) + .5)
IF d * d = rm AND d >= c THEN
PRINT a; b; c; d, a * a; b * b; c * c; d * d
END IF
NEXT
NEXT
NEXT
NEXT n
shows the values of a,b,c and d in ascending order, so the values can be any permuation of these:
0
1 1 1 2 1 1 1 4
1
1 1 1 5 1 1 1 25
1 3 3 3 1 9 9 9
2 2 2 4 4 4 4 16
2
2 2 2 10 4 4 4 100
2 6 6 6 4 36 36 36
4 4 4 8 16 16 16 64
3
4 4 4 20 16 16 16 400
4 12 12 12 16 144 144 144
8 8 8 16 64 64 64 256
4
8 8 8 40 64 64 64 1600
8 24 24 24 64 576 576 576
16 16 16 32 256 256 256 1024
5
16 16 16 80 256 256 256 6400
16 48 48 48 256 2304 2304 2304
32 32 32 64 1024 1024 1024 4096
6
32 32 32 160 1024 1024 1024 25600
32 96 96 96 1024 9216 9216 9216
64 64 64 128 4096 4096 4096 16384
7
64 64 64 320 4096 4096 4096 102400
64 192 192 192 4096 36864 36864 36864
128 128 128 256 16384 16384 16384 65536
8
128 128 128 640 16384 16384 16384 409600
128 384 384 384 16384 147456 147456 147456
256 256 256 512 65536 65536 65536 262144
9
256 256 256 1280 65536 65536 65536 1638400
256 768 768 768 65536 589824 589824 589824
512 512 512 1024 262144 262144 262144 1048576
10
512 512 512 2560 262144 262144 262144 6553600
512 1536 1536 1536 262144 2359296 2359296 2359296
1024 1024 1024 2048 1048576 1048576 1048576 4194304
Other than for n = 0, these are:
2^(n-1), 2^(n-1), 2^(n-1), 5 * 2^(n-1)
2^(n-1), 3 * 2^(n-1), 3 * 2^(n-1), 3 * 2^(n-1)
2^n, 2^n, 2^n, 2^(n+1)
The puzzle does not specify that they be in ascending order, so permutations within each of these also satisfy for a, b, c and d.
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Posted by Charlie
on 2013-08-23 22:55:26 |
computer-guided solution